Shota built a time travel machine, but he can't control the duration of his trip. Each time he uses the machine he has a $0.8$ probability of staying in the alternative time for more than an hour. During the first year of testing, Shota uses his machine $20$ times. Assuming that each trip is equally likely to last for more than an hour, what is the probability that at least one trip will last less than an hour? Round your answer to the nearest hundredth. $P(\text{at least one} < 1\text{ hour})=$
Answer: Strategy In this situation it is much easier to calculate the probability of the event we are looking for (at least one trip lasts less than an hour) by calculating the probability of its complement (every trip lasts at least an hour), and subtracting from $1$. In other words, we can use this strategy: $P(\text{at least one} < 1\text{ hour})=1-P(20\text{ trips at least 1 hour})$ Calculations $\begin{aligned} &\phantom{=}P(\text{at least one} < 1\text{ hour}) \\\\ &=1-P(20\text{ trips at least 1 hour}) \\ \\ &=1-(0.8)^{20} \\ \\ &\approx 1-0.0115 \\ \\ &\approx 0.9885\end{aligned}$ Answer $P(\text{at least one} < 1\text{ hour}) \approx 0.99$